Asymptotic behaviour under iterated random linear transformations (Q1884477)

Let \(X_1,X_2,\dots\) be independent random variables with values in the space \(GL(V)\) of linear transformations of a finite dimensional real vector space \(V\) with identical distributions \(\mu\). Then the product \(Z_i=X_i\cdot X_{i-1}\dots X_1\) corresponds to the application of \(i\) random linear transformations. Let \(G(\mu)\) denote the smallest closed subgroup of \(GL(V)\) containing the support of \(\mu\). The authors study the limit behaviour of \(Z_i\cdot\nu\) when \(\nu\) is a probability measure on \(V\). Let \(B\) be the subspace of \(V\) containing all elements \(v\) such that the \(G(\mu)\)-orbit of \(v\) is bounded in \(V\). Among other results, it is shown that \(Z_i\cdot\nu\) tends to 0 in the vague topology if and only if \(\nu(B)=0\).